Second-order case For $n=2$, by noting $y=x^m$, the ODE provides the indicial equation: \ [\boxed {am^2+ (b-a)m+c=0}\] with discriminant $\boxed {\Delta= (b-a)^2-4ac}$ and where the

In general, given a second order linear equation with the y-term missing y″ + p(t) y′ = g(t), we can solve it by the substitutions u = y′ and u′ = y″ to change the equation to a first order linear

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To solve a linear second order differential equation of the form d2ydx2 + pdydx+ qy = 0 where p and qare constants, we must find the roots of the characteristic equation r2+ pr + q = 0 There are three cases, depending on the discriminant p2 - 4q. When it is positivewe get two real roots, and the solution is y = Aer1x + Ber2x zero See more

Second-Order Ordinary Differential Equation Second Solution If one solution () to a second-order ordinary differential equation (1) is known, the other () may be found using the

I discuss and solve a 2nd order ordinary differential equation that is linear, homogeneous and has constant coefficients. In particular, I solve y'' - 4y' + 4y = 0. The solution method involves